On $p$-groups with automorphism groups related to the exceptional Chevalley groups

Let $\hat G$ be the finite simply connected version of an exceptional Chevalley group, and let $V$ be a nontrivial irreducible module, of minimal dimension, for $\hat G$ over its field of definition. We explore the overgroup structure of $\hat G$ in $\mathrm{GL}(V)$, and the submodule structure of the exterior square (and sometimes the third Lie power) of $V$. When $\hat G$ is defined over a field of odd prime order $p$, this allows us to construct the smallest (with respect to certain properties) $p$-groups $P$ such that the group induced by $\mathrm{Aut}(P)$ on $P/Φ(P)$ is either $\hat G$ or its normaliser in $\mathrm{GL}(V)$.

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